Estudi teòric dels efectes de la solvatació sobre l’espectre UV i IR de la N-metilacetamida.

Dr. Juan Carlos Paniagua Valle

Departament de Química Física

Treball Final de Grau

Theoretical study of the solvation effects on UV and IR spectra of

the

N

-methylacetamide.

Estudi teòric dels efectes de la solvatació sobre l’espectre UV i IR

de la

N

-metilacetamida.

Alejandro Martín Rodríguez

Aquesta obra esta subjecta a la llicència de: Reconeixement–NoComercial-SenseObraDerivada

Els ordinadors són inútils. Només poden donar respostes.

Pablo Picasso

En primer lloc, voldria agrair al Dr. Juan Carlos Paniagua la seva constant dedicació a fer possible aquest treball que tanca la meva etapa com a estudiant de grau de química. Des del primer moment ha estat un plaer estar sota la teva molt valorada supervisió. Espero que en algun moment recuperis tot el temps que has dedicat a que això arribi a bon port i que els teus companys de departament et perdonin per arribar sistemàticament tard a l'hora de dinar cada dimecres.

D'una manera més indirecte però no menys important, voldria agrair a la facultat de química, concretament al professorat que en les seves classes ha permès que tingui una visió més global d'aquesta ciència (i art!) que és la química. Quedeu perdonats de les innumerables hores d'estudi que he hagut de dedicar en aquesta etapa. Han valgut la pena!

Als companys i companyes de trinxera amb els que he compartit moltes anècdotes i totes bones. Les xerrades a mig matí intentant no escaldar-nos la llengua amb el cafè extremadament calent del bar han estat un gran ajut per afrontar tota la feina que tocava fer, tot i que de vegades... Ens despistàvem. Només desitjo que continuem escaldant-nos la llengua junts!

Finalment, agrair a la família tot el recolzament incondicional que m'han donat al llarg de tots aquests anys, els ja passats i els que vindran.

C

ONTENTS

1.

S

UMMARY 3

2.

R

ESUM 5

3.

I

NTRODUCTION 7

3.1. Density functional theory (DFT) 7

3.1.1. The Philosophy 7

3.1.2. Rigorous foundation: Hohenberg-Kohn theorems 8

3.1.2.1. The Hohenberg-Kohn existence theorem 9

3.1.2.2. The Hohenberg-Kohn variational theorem 10

3.1.3. Improving the Hohenberg-Kohn theorems: Levy theorems 10 3.1.3.1. Improving the Hohenberg-Kohn variational theorem 10 3.1.3.2. Improving the Hohenberg-Kohn existence theorem 12

3.1.4. The Kohn-Sham self-consisted field methodology 13

3.2. UV and IR spectra 18

3.2.1. The electromagnetic radiation 18

3.2.2. Historical background: The need of the quantum physics 19

3.2.3. The IR and UV-visible absorption spectra 20

3.2.4. Time-Dependent DFT (TD-DFT) 23

3.3. Solvation models 26

3.3.1. IEF-PCM model 27

3.3.2. SMD model 27

3.4. Density functional dispersion correction 28

4.

O

BJECTIVES 30

5.

C

OMPUTATIONAL DETAILS

30

5.1. Scans and geometry optimization 30

5.2. Solvation 31

6.

C

ONFORMATIONS AND CONFORMERS 32

7.

H-B

ONDING EFFECTS

:

NMA-

CLUSTER 34

8.

IR

SPECTRA

37

9.

UV

SPECTRA

40

10.

C

ONCLUSIONS

45

11.

R

EFERENCES AND NOTES

47

12.

A

CRONYMS

49

A

PPENDICES

Appendix 1: Geometry tables 51

1.

S

UMMARY

This discussion examines the effects of solvation in ultraviolet and infrared spectra of N -methylacetamide at theoretical level. In the first part, the basis of DFT method is explained. In addition, a review of some basic concepts of spectroscopy and a short description of continuum solvation models and Grimme dispersion correction is made. In the second part, the effects of solvation are considered at three different levels: through a continuum description of solvation, using a solute-solvent cluster and using the same cluster embedded in an external continuum. To improve the overview on the effects of solvation on the IR and UV spectra, two types of continuum models are used: IEF-PCM (2003 and 2009 versions) and SMD. All calculations were carried out using two different basis sets: 6-31+G(d,p) and 6-311++G(d,p). In the latter, the dispersion correction proposed by Grimme was added.

Finally, it is analyzed and justified qualitatively the results of both spectra. For the IR spectra, the four fundamental bands amide I, II, III and NH stretching are studied. With respect to ultraviolet absorption, transitions n → π * and π → π * are studied.

Keywords: Time-dependent density functional theory, UV spectroscopy, solvation, IR spectroscopy, DFT, N-methyacetamide, computational chemistry.

2.

R

ESUM

En aquesta discussió s'estudia l'efecte de solvatació en l'espectre infraroig i ultraviolat de la

N-metilacetamida a nivell teòric. En una primera part s'expliquen les bases del mètode DFT. A més, es repassen alguns conceptes bàsics sobre espectroscòpia i es fa una petita introducció als models de solvatació continus i a una correcció de la dispersió proposada per Grimme. En la segona part, es considera l'efecte de la solvatació a tres nivells diferents: mitjançant una descripció continua de la solvatació, utilitzant un clúster solut-solvent i el mateix clúster en solvatació continua. Per tal de millorar la visió sobre l'efecte de la solvatació sobre els espectres IR i UV, s'han emprat dos tipus de models continus: IEF-PCM (versions de 2003 i 2009) i SMD. Tots els càlculs s'han dut a terme mitjançant dos conjunts de bases diferents: 31+G(d,p) i 6-311++G(d,p). En aquest últim, la correcció de la dispersió proposada per Grimme s'afegeix.

Finalment, s'analitzen i justifiquen de manera qualitativa els resultats obtinguts d'ambdós espectres. Per a l'espectre IR, s'estudien les quatre bandes fonamentals amida I, II, III i l'estirament NH. Respecte l'absorció a l'ultraviolat, les transicions n → π* i π → π* són estudiades.

Paraules clau: Time-dependent density functional theory, espectroscòpia UV, solvatació, espectroscòpia IR, DFT, N-metilacetamida, química computacional.

3.

I

NTRODUCTION

In the last years, computational chemistry has become a paramount tool for theoretical chemists. This discipline uses several different methods to describe as far as possible the chemical system. Depending on the system and the objective of the calculation, it would be more useful a specific method than other just because of its way of approaching the real chemical system. The most popular quantum-mechanics-based methods for electronic structure calculations are Hartree-Fock (HF), Møller-Plesset perturbation theory (MPn) and semi-empirical methods among those focused on approximating the wave function of the system, and Kohn-Sham Density Functional Theory (KS-DFT) among those aiming at directly obtaining the electron density. All of them rely on the choice a basis set of functions to expand the molecular orbitals that will describe the electronic structure. There are plenty of different basis sets, most of them based Gaussian type orbitals (GTO), but also Slater type orbitals (STO) are used.

For huge molecules it is necessary to change the strategy because the computational time would be unviable for quantum methods, and then, molecular mechanics (classical mechanics) methods with its different force fields become the best option.

In this work, Density functional theory (DFT) will be used. The basis set used and other considerations would be explained along the document.

3.1.

D

ENSITY FUNCTIONAL THEORY

(DFT)

3.1.1. The philosophy

Most of the methods used to calculate different observables of the system of interest are based on the knowledge of its wave function. This is a tough work, since this function depends on one spin and three spatial coordinates for every electron in the system (assuming Born-Oppenheimer approximation, i.e. fixed nuclear positions). In terms of HF approximation, it is possible to express the state function as a Slater determinant of independent one-electron functions. This wave function can be dramatically improved in quality by removing this rough approach by including a dependence on interelectronic distances in the wave function (R12 methods). It all together seems like the issue is finely performed, but by using wave function

methods, in fact, there is a key disadvantage: The wave function is essentially uninterpretable. It is such a black box that answers accurately when questioned by quantum mechanical operators, but it offers little by way of sparking intuition.

When things start being complicated, it is a wise option to return to the essentials. For instance, why should it be working with a wave function? Can it be working with some physical observable in determining the energy (and other properties) of a molecule? By using what it is already known from quantum mechanics, the Hamiltonian depends on the positions and atomic numbers of the nuclei and the total number of electrons. This immediately suggest that the electron density ρ is a nice observable, because integrated all over the space yields the total number of electrons and it should be obvious that the nuclei positions corresponds to local maxima in the electron density. Though it all, it is one important issue left: The assignment of nuclear atomic numbers. It can be proved that the derivative of the spherically averaged density at the atomic position is proportional to the atomic number.

All the arguments above do not provide any formalism to construct the Hamiltonian operator, solve the Schrödinger's equation, and determine the wave functions and energies eigenvalues. Nevertheless, they suggest that some simplifications might be possible.

3.1.2. Rigorous foundation: Hohenberg-Kohn theorems

Early DFT models were provocative in its simplicity compared to wave-function-based approaches, but failed to describe properly molecular properties, yielding large errors and little impact on chemistry. Even though, the method succeeded in solid-state physics (where the enormous system size puts a premium on simplicity).

One of the simplest models was the uniform electron gas, also known as "jellium". This system is composed of an infinite number of electrons moving in an infinite volume characterized by a uniformly distributed positive charge. This model breaks down for molecules since it ignores the energetic effects associated with correlation and exchange in the calculation of interelectronic repulsion. As result, all molecules are unstable relative to dissociation into their constituent atoms.

In 1964, the theory was finally rigorously founded. Hohenberg and Kohn proved two key theorems to establish a variational principle for DFT calculations and then, legitimate the method as a quantum mechanical methodology.

3.1.2.1. The Hohenberg-Kohn existence theorem

The first theorem states the ground-state density determines the external potential. This potential defines the interaction between electrons and nuclei.

The proof of the existence theorem proceeds via reductio ad absurdum: Let us assume that two different external potentials can be obtained from the same non-degenerated ground-state density ρ0. This two potentials, va and vb and its different Hamiltonians in which they appear Ha

and Hb will be associated to a ground-stated wave function ψ0 and its associated eigenvalue E0.

From the variational theorem of molecular orbital theory its known that

b a b

a

H

E

_0,

<

ψ

_0,

|

|

ψ

_0, (3.1)

And can be rewrite as

b b b a b a

H

H

H

E

0,

<

ψ

0,

|

−

+

|

ψ

0, b b b b b a b a

H

H

H

E

_0,

<

ψ

_0,

|

−

|

ψ

_0,

+

ψ

_0,

|

|

ψ

_0, b b b a b a

H

H

E

E

_0,

<

ψ

_0,

|

−

|

ψ

_0,

+

_0, (3.2)

Since the potentials v are one-electron operators, the integral in the last line can be expressed in terms of the ground-state density

b b a a

r

r

r

dr

E

E

_0,

<

∫

[

υ

(

)

−

υ

(

)

]

ρ

₀

(

)

+

_0, (3.3) a a b b

r

r

r

dr

E

E

_0,

<

∫

[

υ

(

)

−

υ

(

)

]

ρ

₀

(

)

+

_0. (3.4) As there is no distinction made between a and b, it is possible to switch the indices in Eq. (3.3) to obtain Eq. (3.4). Now, adding inequalities

dr

r

r

r

dr

r

r

r

E

E

_0,a

+

_0,b

<

∫

[

υ

_a

(

)

−

υ

_b

(

)

]

ρ

₀

(

)

+

∫

[

υ

_a

(

)

−

υ

_b

(

)

]

ρ

₀

(

)

a b

E

E

_0,

+

_0,

+

a b b b a a b a

E

r

r

r

r

r

dr

E

E

E

_0,

+

_0,

<

∫

[

υ

(

)

−

υ

(

)

−

υ

(

)

+

υ

(

)]

ρ

₀

(

)

+

_0,

+

_0, a b b a

E

E

E

E

_0,

+

_0,

<

_0,

+

_0, (3.5)

The expression achieved left an impossible result, which must indicate that the initial assumption is incorrect. So, a non-degenerated ground-state density must determine the external potential, thus, the Hamiltonian and finally, the wave-function.

Note moreover that the Hamiltonian determines also all excited-state wave-functions as well. But. Is it possible to calculate any excited state? Gunnanrsson and Lundqvist, using group

theory, proved that only the lowest energy (non-degenerated) state within each irreducible representation of the molecular point group can be calculated. That means, for instance, the densities of the singlet Ag and triplet B1u states of p-benzyne are well defined, but nothing can

be said about triplet A' state of N-protonated 2,5-didehydropyridine, since there is a lower energy singlet A'. The development of DFT formalism to handle arbitrary excited states remains a subject of active research.

3.1.2.2. The Hohenberg-Kohn variational theorem

The first theorem is an existence theorem, but gives no information about how to predict the density of a system. Hohenberg and Kohn showed in a second theorem that, as with MO theory, the density obeys a variational principle.

Let us assume a well-behaved candidate density that determines ψcand and its Hamiltonian.

Invocating the variational theorem

0

|

|

H

_{cand cand}

E

_cand

E

cand

ψ

=

≥

ψ

(3.6)

So, in principle, it is possible to keep choosing different densities and those that provide lower energies, are closer to correct. But first, there is no formalism to improve the candidate densities rationally, and second, it is not avoided to solve Schrödinger equation - It is well known to do that. Nevertheless, most of practical implementations requires to calculate a wave-function for a simpler hypothetical system.

Such an approach to reach an optimal density was propose in 1965 through a self-consistent field procedure by Kohn and Sham. This SCF will be discussed in section 3.1.4.

3.1.3. Improving the Hohenberg-Kohn theorems: Levy theorems

In 1979, M. Levy demonstrated Hohenberg and Kohn theorems by using a formulation based on restricted minimizations. This new formulation expands the original theorems and connects them closely with the variational theorem.

3.1.3.1. Improving the Hohenberg-Kohn variational theorem

For a n-electronic molecule, it is possible to express the Hamiltonian as

(3.7)

∑∑

∑∑

∑

= = − = > =

−

+

+

∇

−

=

n i N A iA A n i n i j ij n i i

r

Z

r

H

1 1 1 1 1

1

2

ˆ

Where the first term is the kinetic energy operator (Tel), the second is the interelectronic

repulsion operator (Vel) and the last term is the attractive potential between the electrons and

the nuclei(Vnuc-el).

The monoelectronic density of a wave-function ψ and the wave-function itself is related through the defined integral

n n

d

r

d

d

r

d

d

r

d

r

,

ω

₁

,

r

₂

,

ω

₂

,...,

r

,

ω

×

(3.8)

Eq. (3.8) shows that different wave-functions could yield to the same density. This permits to express the variational theorem (Eq. (3.6)) as a two-steps minimization:

1. For a target density, the wave-functions ψρ with that density and energy

minimized are chosen.

2. From the set generated before, the minimal energy function is chosen. After all, the process can be expressed as

(3.9) Thus, the connection with the variational theorem is already shown. If it were possible to know the explicit form of E[ρ], it would be possible to minimize directly the functional, and hence, there would be no need of calculating ψρ.

Yet another highlight: If it combines Eq. (3.7) and Eq. (3.9)

[ ]

min

(

)

min min min

|

ˆ

|

_{ρ ρ ρ} ρ

ψ

ψ

ψ

ψ

ρ

T

el

V

el

V

nuc el

E

=

+

+

₋ (3.10)

As the former term uniquely depends on the number of electrons, it is the same functional for every system with the same number of electrons. So, it can be redefined as the universal functional F[ρ].

[ ]

min

(

_ˆ

)

min

|

_ρ ρ

ψ

ψ

ρ

T

el

V

el

F

=

+

(3.11)

The minimization of this functional is perfectly equivalent as minimizing Eq. (3.9) as the latter term of Eq. (3.10) yields to the same value for different wave-functions with the same density. If the universal functional were perfectly known, the sum of the latter term to the former would be enough to obtain the functional E[ρ].

[ ]

V

nuc el

F

E

0

=

ρ

+

₋ (3.12)

∫ ∫ ∫ ∫ ∫

=

1 2 2 2 2 2 1

,

,

,...,

,

)

|

,

(

|

···

)

(

ω ω ω

ψ

ω

ω

ω

ρ

r rn n n n

r

r

r

n

r

r r

r

r

r

r

[ ]

0 0

min

_ρ

min

_ψ

ψ

ρ

|

ˆ

ψ

ρ

ρ

ρ

E

H

E



=











=

Thus, Levy proved and improved the Hohenberg-Kohn variational theorem. The original theorem was restricted to non-degenerated ground-state. This new formulation expands the theory to degenerated ground-states.

Hohenberg-Kohn and Levy formulation explain that if the wave-function is known, the density is also known, but not vice versa. As it is able only to know a wave-function that has the same density, the properties do not have to be the expected ones. This would be a big limitation, but actually it is not a big issue since the energy is determined by Eq. (3.12) and the other properties can be easily related with the density.

3.1.3.2. Improving the Hohenberg-Kohn existence theorem

As in the original enunciation, this theorem proves that the external potential is uniquely defined given a monoelectronic density. In Levy's formulation, the Hohenberg-Kohn theorem is expanded to degenerated ground-states: any degenerated density of the ground-state yields to the same external potential. The external potential is defined by

(3.13) An intuitive justification of this expression: If the monoelectronic density of the ground-state (or any excited state) is known, is possible to calculate the number of electrons, which determinates the operators Tel and Vel. The positions of the nuclei are defined where the density

has a peak, which slope is dependent of the charge of the nuclei. Hence, it is possible to calculate the operator Vnuc-el.

Using the definition of functional derivative, it is possible to prove Eq. (3.13)

(3.14) where ε is a real parameter and g(r) is an arbitrary function of r. For proving Eq. (3.13), ρ is defined as

)

(

0

g

r

r

ε

ρ

ρ

=

+

(3.15)

where g is any function that accomplishes the normalization of ρ, that is

(3.16) Combining Eq. (3.16), Eq. (3.12) and Eq. (3.9).

[ ]

0

)

(

)

(

ρ ρ

δρ

ρ

δ

=













−

=

r

F

r

v

_ne

r

r

[

]

_∫

[ ]

ℜ = =













=













∂

+

∂

3 0

)

(

)

(

)

(

0 0

_g

_r

_d

_r

r

F

r

g

F

r

r

r

r

ρ ρ ε

δρ

ρ

δ

ε

ε

ρ

0

)

(

3

=

∫

_ℜ

g

r

r

d

r

r

(3.17) As ρ0 is the minima of E[ρ] (first theorem), ε must equal 0. Also, ε=0 must be a stationary

point

(3.18) Hence, using Eq. (3.14)

(3.19) Eq. (3.19) shows that for any g(r) function, the term in braces must be constant.

(3.20) Allowing this constant to be zero, Eq. (3.13) is achieved.

3.1.4. The Kohn-Sham self-consistent field methodology

The main blockage to develop a DFT method is the lack of awareness of the universal functional. W. Kohn and L. J. Sham propose in 1965 a strategy to approach the external potential. They used a hypothetic n-electron independent system under a multiplicative external potential vKS_{(r) with an associated ground-state monoelectronic density ρ0KS equal to the density}

of the real system.

This strategy works under the Kohn-Sham conjecture: If the Kohn-Sham system exist, then, the density of the real system is representable. If just one Kohn-Sham system is enough to perform the target density of the real system, that system is called pure-state representable. If a weighted sum of monoelectronic densities is needed to perform the real system, then the system is Ensemble representable.

The Hamiltonian of a Kohn-Sham system is defined as

(3.21)

[ ]

{

[

]

[

]

}

[

]

[

]

{

F

g

r

v

r

g

r

d

r

}

r

g

V

r

g

F

E

E

ne el nuc

r

r

r

r

r

r

)

(

)

(

)

(

min

)

(

)

(

min

min

0 0 0 0 0 3

ρ

ε

ε

ρ

ε

ρ

ε

ρ

ρ

ε ε ρ

+

+

+

=

+

+

+

=

=

∫

_ℜ −

[

]

[

]

{

}

0

)

(

)

(

)

(

0 0 0 3

=

















∂

+

+

+

∂

= ℜ

∫

ε

ε

ε

ρ

ε

ρ

g

r

v

r

g

r

d

r

F

ne

r

r

r

r

[ ]

₍

₎

₍

₎

₀

)

(

3 0

=

















+













∫

_ℜ =

r

d

r

g

r

v

r

F

ne

r

r

r

r

ρ ρ

δρ

ρ

δ

[ ]

_v

_r

_c

r

F

ne

=

+













=

)

(

)

(

0

r

r

ρ ρ

δρ

ρ

δ

)

(

ˆ

)

(

2

ˆ

ˆ

ˆ

1 1 i n i KS n i i KS ne KS ne el KS

r

h

r

v

V

T

H

∑

r

∑

r

= =

=













−

∇

₊

=

+

=

where the operator Vel does not appear as long as the electrons have no interaction

between them and hKS _{is the monoelectronic Kohn-Sham operator.}

As in the Hartree-Fock method, the separability of HKS _{allows to express the ground-state}

wave-function as a Slater determinant built up with n eigenspinorbitals of the KS operator with lower energy. (3.22) with

)

(

)

(

)

(

ˆ

r

_ψ

_ω

r

_ε

_ψ

KS

_ω

r

i KS i KS i i KS

r

h

=

(3.23) and energy (3.24) If the Kohn-Sham system ground-state is degenerated, there is no problem as long as all Slater determinants have the same electronic densities (i.e. non-degenerated states or one unpaired electron in a non-degenerated orbital). But if the Kohn-Sham system has, for instance, a 1s2_2s2_{2px2 configuration, this would have the same energy as a 1s}2_2s2_{2px12py1 system (both}

are degenerated), but not the same density. Hence, it cannot be know which density corresponds to the real system. Although several strategies are studied to approach this cases, it will be focused on the degenerated cases and those with one despaired electron in a non-degenerated orbital.

Both theorems apply to any n-electronic system; thus, we can apply them to a Kohn-Sham system. The universal functional F[ρ] to be minimized then reduces to

[ ]

[ ]

min

_ˆ

min

|

_ρ ρ

ρ

ρ

=

=

Φ

el

Φ

KS KS

T

T

F

(3.25)

where the wave-function that minimize the functional can be expressed as a Slater determinant. The previous equation leads to the modification of Eq. (3.13)

(3.26) Now it is time to impose ρ0 to be the density of the real system. Nevertheless, the

wave-function that minimizes F[ρ] is not the same that minimizes FKS_{[ρ]. An attempt to do so, is to}

)

(

)...

(

!

1

1 1 n KS n KS KS

n

ψ

ω

ψ

ω

=

Φ

∑

=

=

n i KS i KS

E

1

ε

[ ]

0

)

(

)

(

ρ ρ

δρ

ρ

δ

=













−

=

r

T

r

v

KS KS

r

r

adopt TKS_{[ρ] as an approach to T[ρ] and the classical columbic repulsion J[ρ] as an approach to}

Vel[ρ]

(3.27) where the 1/2 factor is included so as not double count every pair of electrons. To take into account the error produced in such approximation, a new functional known as exchange-correlation functional Exc[ρ] is introduced.

[ ] [ ]

ρ

T

ρ

T

[ ]

ρ

V

[ ] [ ]

ρ

J

ρ

E

el

KS

xc

=

−

+

−

(3.28)

This new functional is also universal as long as all the terms appearing in its definition are universal. The Exc functional has to gather the following considerations even though both

approximations are fair enough and the functional is small:

The lack of electronic correlation in the Slater determinant used for calculating TKS_[ρ].

• The lack of electronic correlation in J[ρ] calculation.

• Exchange effects

• The self-repulsion introduced by the use of J[ρ].

The exchange effects are, by far, the largest of them all. Through an analogous deduction used for Eq. (3.13), the correlation-exchange potential is define as

(3.29) and then, combining Eq. (3.29), Eq. (3.13) and Eq. (3.26)

(3.30) Sadly, the real monoelectronic density is still unknown. But now (and assuming an approach to vxc), a self-consistent field calculation is used to determine vKS (figure 3.1).

[ ]

1 2 12 2 1 1 2

)

(

)

(

2

1

r

d

r

d

r

r

r

J

r r

r

r

r

r

r r

∫ ∫

=

ρ

ρ

ρ

[ ]

0

)

(

)

(

ρ ρ

δρ

ρ

δ

=













=

r

E

r

v

xc xc

r

r

)

(

)

(

)

(

)

(

_{2 1} 12 2 0 1 1 2

r

v

r

d

r

r

r

v

r

v

_xc r ne KS

r

r

r

r

r

r

+

+

=

_∫

ρ

Figure 3.1. Kohn-Sham Self-consistent field

Choose a basis set(s)

Choose a molecular geometry

Compute and store all overlap

and one-electron integrals Guess initial density matrix P

(0)

Optimize molecular geometry?

Construct KS operator and construct density matrix from KS MOs.

New density P(n) _{is similar}

enough to old density P(n-1)_?

Replace P(n-1) _{with P}(n)

Solve Eq. (3.30)

Output data for unoptimized geometry

Current geometry satisfies the optimization criteria?

Choose new geometry

according to optimization algorithm.

Output data for optimized geometry yes no yes yes no

Several approaches to vxc are briefly explained in the next headland.

The density from the KS occupied orbitals is calculated through

(3.31) where ni is the occupation number of the orbital i. With Eq. (3.31) it is possible to build up a

gridbox with a defined density for a given point of space.

3.1.4.1. Exchange-correlation functionals

In this subsection it will be enumerated some exchange-correlation functionals to use in the previous SCF. In many of this functionals empirical parameters appear.

Usually, the functional dependence of Exc is expressed as an integral of the electron density

times an energy density εxc. The Exc is obtained with

[ ]

r

r

[ ]

r

dr

E

_xc

ρ

(

)

=

∫

ρ

(

)

ε

_xc

ρ

(

)

(3.32)

Another convention expresses the electron density in terms of an effective radius such that exactly one electron would be contained within the sphere defined by that radius were it to have the same density throughout as in its centre

(3.33) For open shell systems the spin density at any position is typically expressed in terms of ζ, the normalized spin polarization

(3.34) This considerations lead to what is known as “pure” functionals:

• Local density approximation (LDA): The value of εxc at some position r can be

computed exclusively from the value of ρ and is single-valued at every position.

• Local spin density approximation (LSDA): LDA is extended to a spin-polarized formalism.

• Generalized gradient approximation (GGA): One obvious way to improve the correlation functional is to make it depend not only on the local value of the

2 1 0

n

|

i

(

r

)

|

oc i i

r

ψ

ρ

∑

=

=

3 / 1

)

(

4

3

)

(

_











=

r

r

r

_S

πρ

)

(

)

(

)

(

)

(

r

r

r

r

ρ

ρ

ρ

ζ

=

α

−

β

density, but on the extent to which the density is locally changing. So, LDA is gradient corrected.

• meta-GGA (m-GGA): The second derivative is added to GGA.

There are other types of pure functionals, but the previous types are the most representative.

Another class is that of the so-called hybrid functionals, which mix pure functionals with the Hartree-Fock exchange energy to improve the results of both individual methods. One of the most popular within this class, the B3LYP functional is used in the calculations of this work and corresponds to LYCP c LSDA c B x HF x LSDA x LYP B xc

a

E

aE

b

E

c

E

cE

E

3

=

(

1

−

)

+

+

∆

+

(

1

−

)

+

(3.35)

where a, b and c take the values 0.20, 0.72 and 0.81.

3.2.

UV

AND

IR

SPECTRA

In this section it will be provided some qualitative basis of absorption spectroscopy, specially for ultra-violet (UV) and infrared (IR) wavelengths, needed to understand the results of the present work. Perhaps this part is not worth it for an specialist reader, but it may be a good reminder for this topic.

3.2.1. The electromagnetic radiation

A classical view of the electromagnetic radiation is to consider that it is produced when charged particles are accelerated by forces acting on them. This perturbation propagates through the space carrying radiant energy at speed of light in vacuum. As an electromagnetic wave, it has both electric (E) and magnetic (B) oscillating fields perpendicular to each other and also to the direction of propagation (k). James Clerk Maxwell demonstrate through his equations that electricity, magnetism and light are because of the same phenomena: the electromagnetic field.

The λ shown in figure (3.2) is called wavelength, the length between two points of the same phase. Through this magnitude, the electromagnetic waves can be classified. This classification is known as electromagnetic spectrum (figure 3.3).

Figure 3.2. Electromagnetic wave.

(Emmanuel Boutet, 03/06/14 via Wikimedia commons, Creative commons attribution)

Figure 3.3. Electromagnetic spectrum.

(Inductiveload, 03/06/14 via Wikimedia commons, Creative commons attribution)

3.2.2. Historical background: The need of the quantum physics

As an electromagnetic waves carry energy, this energy can be absorbed or emitted by a molecule (and by extension, a macroscopic body). In late s.XIX, the emission spectra of an incandescent body was an object of study. The Stefan-Boltzmann law predicted the total intensity of radiation given a temperature, and the Wien Law predicted, given a temperature, which wavelength was the most emitted.

All seemed well until Rayleigh and Jeans predicted that, for classical physics, a body with a non-zero temperature should emit ultra-violet radiation. Even X rays and gamma rays. In fact, the human body at 37ºC would shine in the darkness! This unbelievable (but true at classical physics level) prediction is known as Rayleigh-Jeans Catastrophe or Ultra-violet catastrophe.

Luckily, in 1900, the German physicist Max Planck achieved the solution to the ultra-violet catastrophe: the energy exchange between the electromagnetic wave and the matter should be

quantized. The Planck law performs correctly the radiation emitted by an incandescent body. The main idea was that the energy can only be exchanged with the matter in packs of energy defined by

υ

h

E

=

(3.36)

where h is the Planck constant and has a value of 6,626·10-34 _{J·s and ν is the frequency of}

the wave or oscillation. This simple equation represents the birth of quantum physics. In 1905, Albert Einstein extended Planck's hypothesis to explain the photoelectric effect. From Planck and Einstein works, Louis De Broglie developed the wave-particle duality. This theory proposes that elementary particles are not only particles, but also waves. The elemental particle associated to the electromagnetic waves of light are photons. The energy of a photon of a given wavelength is defined by Eq. (3.36).

So, actually, the interpretation of an absorption can be approximated to the collision of a photon and the molecule and/or an electron. In this collision, the energy is exchanged and the system suffers an electronic change (usually in the visible or UV range) or a vibrational change (in the IR range).

3.2.3. The IR and UV-visible absorption spectra

In fact, a molecule can absorb a wide range of the electromagnetic spectra, but only in some specific wavelengths. This quantized wavelengths are determined by the characteristics of the molecule.

The changes in the vibrational energy of the molecule produce absorption bands in the IR range, from the visible limit to about 100 µm.

A molecule can vibrate in many ways and each way of vibrating is called vibration mode. The number of vibration modes for linear molecules (for instance CO2) is 3N-5, where N is the

number of atoms. A non-linear molecule has 3N-6 different vibrations. The classical vibrations that can be seen in a -CH2- group and other groups are shown in Figure 3.4.

Nevertheless, there are some special effects: in some cases, overtone bands are shown. This bands are because of the absorption of a photon that leads to a double excited vibrational state (i.e. the vibrational state goes from fundamental state to the second excited state). Such bands appear at twice the energy of the fundamental. Furthermore, second or third overtones could be seen, but the intensity of those bands is smaller.

Also, the so-called combination modes involves more than one normal mode. This phenomenon arises when the energy of the bands implied are similar in energy.

The UV-visible range of absorption produces a change in the electronic configuration of a molecule. When an electron is excited by a photon, this electron gain energy enough to promote to a higher energy orbital. In fact, a given electron will only interact with a photon that has the exact energy between the fundamental electronic state of the molecule and n the excited state. Where n is the first, second and so excited states. Actually, same idea is perfectly applicable to the IR spectra and other absorption spectra.

Figure 3.4. Classical vibrations of a -CH2- group. In a complex molecule a mix of them can be seen. The energy of that photon can be approached as the energy difference between the orbital where the electron actually is and the orbital to which the electron will be promoted. In spite of this approximation, another fact must be considered: when an electron is promoted, the whole repulsion between electrons is affected. Thus, it is necessary to take into account the variation of the repulsion once the promotion is completed. This concept yields to excitation energies around the energy difference between the orbitals implied, but can be greater or lower than the

approached. On the other hand, molecular geometry can change upon an electronic transition. If this is not taken into account, the energy calculated is the so-called vertical transition excitation energy. After all, even if a photon has the fine energy to promote either an electron or a vibrational mode, the transition must accomplish what is known as selection rules. The more rules a transition satisfy, the greater the intensity of that transition will be.

For an electronic transition:

• The transition must be allowed by electric dipole

• The transition must be allowed by magnetic dipole

• The transition must keep the spin of the system

For a vibrational transition only apply the dipole moment rules as the spin is kept the same for granted.

To summarise this section, it was explained what the radiation is, how the radiation interacts with matter and what requirements are needed to make this happen. Nevertheless, a very important final issue left: if the energy of an photon must be exactly the difference between states either in vibrational or electronic transitions, why a band is shown when an spectrum is performed?

Figure 3.5 illustrates the Frank-Condon principle. This principle says that a transition is more probable if there is no change in the atomic positions. The electronic transitions are essentially instantaneous in comparison to the movement of the nuclei. Hence, if a molecule promotes an electron to a new electronic level, the new vibrational level must have a compatible geometry with the previous one. The blue line shows a vertical transition from the fundamental electronic state (E0) and fundamental vibrational state (v=0) to the first electronic state (E1) and second

excited vibrational state (v=2). So, under Franck-Condon principle, an electronic excitation will carry an intrinsic change on the vibrational level, either to a greater or a lower level. Adding that given a temperature, a molecule will occupy a certain distribution of the firsts vibrational levels (if the energetic difference between levels is low), it is possible to understand why, although the energy is quantized, not a line but a band is shown when a UV-visible spectrum is performed.

As an electronic state has some vibrational levels associated, a vibrational level has some rotational levels associated. An analogous idea can be used to understand why a IR-spectrum produce bands: Given a temperature, a molecule will occupy different rotational levels in a

certain distribution so, for just one excitation, a band will be shown. Once rotational excitations are reached, quantum nature can be seen: a pure rotational spectrum shows, finally, lines.

Figure 3.5. A Franck-Condon diagram.

(Mark M. Somoza, 04/06/13 via Wikimedia commons, Creative commons attribution)

3.2.4. Time-Dependent DFT (TD-DFT)

Now, it is time to focus on how to compute an electronic absorption spectrum. An intuitive way of deal with this issue is to calculate the ground-state energy and the target excited state energy and then, calculate the energy difference. This method actually works, but it is impossible to handle if the objective is to study the whole spectrum. Furthermore, it was already shown in section 3.1.4 that DFT can only know for sure the energy of the lowest non-degenerated excited state of an specific irreducible representation. Hence, only a few bands could be performed with DFT.

A first approximation to calculate the excitation energies is to use the differences between the ground-state Kohn-Sham eigenvalues. The Kohn-Sham orbitalic energies have no direct physical meaning (with the exception of the highest occupied one), but sometimes are used to interpret experimental observations.

More elaborated approaches are

• To restrict the variational principle to wave-functions of an specific symmetry.

• The so-called generalized adiabatic connection Kohn-Sham formalism, where each state of the many-body system is adiabatically connected to a Kohn-Sham state.

• Searching for local extreme of the ground-state energy functional.

• Calculating multiple energies by a Hartree-Fock-Slater method.

• Ensemble DFT, where energy is obtained through an ensemble of the ground state with some excited states.

Alternatively, the excitation energies can be obtained by knowing how the system responds to a small time-dependent perturbation, and can therefore be readily extracted from a TD-DFT calculation. The Time-dependent DFT theory is an extension of DFT to deal with time-dependent systems. This theory stands in virtue of the Runge-Gross theorems. These theorems are completely analogous to H-K/Levy theorems but, this time, the density functionals (and the density itself) are time dependent.

Within TD-DFT, two different regimes can be distinguished:

1. For a strong external potential, a full solution of the TD-DFT Kohn-Sham equations (Eq. (3.23) and Eq. (3.31)) is needed.

2. If the external time-dependent potential is small, the solution of the Kohn-Sham equations can be avoided and the linear response theory can be used.

The former assumption is necessary for intense laser fields (good to remember that the external field can also hold this kind of external interactions). The latter will be used to derivate a self-consisted field to achieve the excitation energies of a target system. Here is presented a simplified derivation of the SCF. A more extended derivation of Runge-Gross theorems, the exchange-correlation TD-functionals (analogous to TD-functionals) and the whole methodology can be consulted in ref. 5 and 6.

So, a change on the density when the system is perturbed by a small change of the external potential is described by

∑ ∫

=

' ' '

(

,

'

,

)

(

'

,

)

'

)

,

(

σ σσ σ σ

ω

χ

ω

δυ

ω

δ

n

r

dr

r

r

ext

r

(3.37)

where χ function is the so-called linear density response function, δvextσ' is the perturbation

on the system. This equation is good enough to evaluate χ for a linear excitation spectrum, but it is easily extended to study nonlinear responses or cases where the external field is not a small perturbation.

Following a similar strategy used to elaborate a SCF, Eq. (3.37) can be used for a Kohn-Sham system. In terms of the unperturbed stationary Kohn-Kohn-Sham orbitals, it reads:

(3.38) where ψ and ε are Kohn-Sham orbitals and its eigenvalues, f is the occupation number of the jth/ith orbital and η is a positive infinitesimal. Using Eq. (3.30) adapted to time dependence we can obtain the linear change of the Kohn-Sham potential

(3.39) The Fourier transform of the xc functional has been introduced

(3.40) Combining Eq. (3.37) for a Kohn-Sham system and Eq. (3.39)

(3.41) As the Kohn-Sham system and the real system must have the same change on the density, combining time again Eq. (3.30), but this time for the real system

(3.42) Eq. (3.42) is a formally exact representation of the density of the interacting system. Nevertheless, the fxc functional is not known and an approximation must be used. Different

∑

∞

−

₋

₋

₊

=

jk j k k k j j j k KS

i

r

r

r

r

f

f

r

r

η

ε

ε

ω

ψ

ψ

ψ

ψ

δ

ω

χ

σ σ σ σ σ σ σ σ σσ σσ

)

(

)

(

)

'

(

)

'

(

)

(

)

(

)

,

'

,

(

* * ' '

∫

+

∑ ∫

−

+

=

' '

(

,

'

,

)

(

'

,

)

'

|'

|

)

,

'

(

'

)

,

(

)

,

(

σ σσ σ σ

ω

δυ

ω

δ

ω

ω

δ

ω

δυ

dr

f

r

r

n

r

r

r

r

n

dr

r

r

_{ext xc} KS

[

]

[

]

)

'

,

'

(

)

,

(

,

)

'

,

'

,

(

,

' '

t

r

n

t

r

n

n

t

t

r

r

n

n

f

xc xc σ σ σσ ↑ ↓

−

=

δυ

_δ

↑ ↓

+







−

+

=

∑ ∫

_∫

τ στ τ σ

ω

χ

ω

δυ

ω

δ

ω

δ

|

'

|

)

,

(

)

,

'

(

)

,

'

,

(

'

)

,

(

x

r

x

n

dx

r

r

r

dr

r

n

ext KS







+

_{∑ ∫}

' '

(

'

,

,

)

(

,

)

τ ττ

ω

δ

ω

n

x

x

r

dxf

_xc

∑ ∫ ∫

×

+

=

' ' '

(

,

'

,

)

(

,

'

,

)

'

(

,

,

)

ττ στ σσ σσ

ω

χ

ω

χ

ω

χ

r

r

KS

r

r

dx

dx

r

x

)

,

'

,

'

(

)

,

'

,

(

|'

|

1

' ' '

ω

χ

τσ

ω

ττ

x

x

x

r

f

x

x

KS xc













+

−

×

approaches are available. The functionals seen in section 3.1.4.1 have their time-dependent relatives and also other functionals not explored in this work can be used.

Once the exchange-correlation functional is approached, a self-consistent field is ready to calculate χ of the interacting system through Eq. (3.42).

3.3.

S

OLVATION MODELS

In this section will be provided some ideas of how to model the solvation. Unlike previous sections the basic idea of this modelling will be explained. An exhaustive discussion of this topic can be consulted in ref 7 and 8.

Here it will be presented two methods, both are polarizable continuum models. PCM models do not represent the solvent explicitly but rather as a dielectric medium with surface tension at the solute-solvent boundary. This type modelling tries to avoid the unviable calculation of explicit solvent molecules around the solute. In the calculations of this work SMD (solvent model density) and IEF-PCM (integral equation formalism-polarizable continuum model) have been used.

The solvent is usually characterized by some or (as SMD) all of these parameters: The main parameter, the dielectric constant. Other parameters are refractive index, bulk surface tension and acidity and basicity parameters.

Once the solvent is parameterized a paramount issue is how is described the solvent-solute interaction, and this issue is mainly what makes the difference between different methods. Nevertheless, a common idea is to describe a cavity were the solute fits in. Out of this cavity, the continuum solvent. The cavity is performed through several criteria depending on the method, and do not must be "molecule shaped", as intuition may predict. For example, geometry and atomic numbers can be used to do so. However, the most physical definition of a solute cavity would surely depend on more details of the solute than just its geometry and the atomic numbers of its constituent atoms, and in fact some solvation models use cavities in which the radii depend on charges and/or atomic environments or the positions of isodensity contours. A precise definition of a solute cavity is uncertain, and actually the very concept of a solute cavity is an approximate one.

From a chemical reasoning, closest molecules of solvent experiments a stronger interaction with the solute (first solvation shell) and further shells are essentially bulk solvent. That means

that geometry of first solvation shell molecules will suffer a considerable change (also the geometry of the solute). Obviously, the solvent-solute interaction becomes stronger as long as the solvent becomes more polar. So, the continuum model must perform this consideration.

As the first solvation shell will always be the most important interaction, this methods focus on the solute-solvent boundary. This important issue is the most difficult to deal with, because as said before, solute cavity is an approximate concept, and several approximations and/or imprecise description of this interaction are granted.

Experimental parameters are used to fit some equations. These parameters are extracted from a set of well-chosen compounds, depending on which parameter is going to be fitted.

3.3.1. IEF-PCM model

Let us discuss some of the considerations taken into account in this method. As said a few lines before, this model approximate the solvent as a continuum medium, characterized by its dielectric constant ε, interacting with the solute molecule.

The solute can be performed in a quantum mechanical level or as a classical collection of point charges and is fitted in a cavity that host the solute, thus defining the solute-solvent interface as the cavity surface Г. Mutual polarization of the solute and the continuum can be recast using boundary conditions. In fact, the cavity surface need to be discretized to carry out the required numerical surface integrations. Nevertheless this process leads to discontinuities, singularities and to an energy functional which is not smooth with respect to the parameters involved in its definition.

A big advantage of this methodology is that represents a very good compromise between accuracy, formal complexity, applicability and widespread availability.

3.3.2. SMD model

Essentially, SMD follows the philosophy of the PCM methods. The charge distribution of the solute (charge density) induces polarization in the surrounding dielectric medium, and the self-consistently determined interaction between the solute charge distribution and the electric polarization field of the solvent, when adjusted for the energetic cost of polarizing the solute and the solvent, constitutes what is called the electrostatic (or bulk electrostatic) contribution to the free energy of solvation. The electric potential of the solute, called the reaction field, equals to the total potential minus the electrostatic potential of the gas-phase solute molecule.

The total electric field satisfies the nonhomogeneous Poisson equation (NPE) for electrostatics. Therefore, the reaction field can be obtained self-consistently by numerical integration of the NPE coupled to the quantum mechanical electron density of the solute molecule.

Sadly, current methods based on the NPE have uncertainties due to the definition, size and shape of the solute cavity, the portion of the solute charge that may lie outside the cavity and the assumed way in which the permittivity changes at and near the solute-solvent boundary.

But what are the differences between using IEF-PCM or SMD model? SMD is a newer methodology and have some advantages respect of IEF-PCM. SMD parameters do not depend on charges or hybridization states, thus there are no discontinuities when the model is applied along reaction paths. Also adds a missing non-electrostatic term to the basic electrostatic picture. Furthermore, SMD is more versatile, because can be used in more free and commercial electronic structure packages and different chemical system yielding no more (nor less) accurate results as other PCM models.

3.4.

D

ENSITY FUNCTIONAL DISPERSION CORRECTION

As for solvation modelling, this section will uniquely provide some considerations of a dispersion correction for DFT. Some methods, within them DFT, fail to describe with some accuracy the weak but physically and chemically very important London dispersion interactions. One could obviate these kind of interactions, but it would suppose a mistake, specially for non-polar molecules and solvents because, even weak, they can make the difference.

The dispersion correction used in this work was published by Stefan Grimme et al. (ref. 9). This method actually can be considered robust as it has been tested thoroughly and applied successfully now on thousands of different systems including inter- and intramolecular cases ranging from rare gas dimers to huge graphene sheets. Also has very nice properties of minor numerical complexity.

Grimme and co-workers achieved almost ab initio fashion consistent parameters for the entire set of chemically relevant elements with a minimal of empiricism. Nevertheless, the main drawback of this method is that is not dependent on and does not affect the electronic structure

in order to keep the numerical complexity low. This restriction ultimately limits the accuracy, in particular, for chemically unusual cases.

Moreover, other dispersion corrections are usually designed for a specific type of functional. Grimme's correction was designed as a correction for common functionals (as B3LYP, PBE or TPSS) that may be not optimal for non-covalent interactions but perform well for other important properties.

Compared to other corrections and previous versions, the current version (2010) has the following properties and advantages:

• Less empirical

• Asymptotically correct with all density functionals for finite systems (molecules) or non-metallic infinite systems (i.e. performs correctly the long-range interactions). It gives the almost exact dispersion energy for a gas of weakly interacting neutral atoms and smoothly interpolates to molecular regions.

• Good description of all chemically relevant elements (nuclear charge Z=1-94).

• Atom pair-specific dispersion coefficients and cutoff radii (interatomic distance region where the dispersion energy is decreasing) are explicit computed.

• New fractional coordination number (geometry) dependent dispersion coefficients are used that do not rely on atom connectivity information (differentiable energy expression).

• Similar of better accuracy for light molecules and a strongly improved description of metallic and heavier systems.

About empiricism, this method only requires adjustment of two global parameters for each density functional.

The overall of the dispersion correction gives a decrease of 15-40% in the mean absolute deviations compared to the previous version and spectacular improvements are found for a tripeptide-folding model and all tested metallic systems, were London interactions become paramount.

4.

O

BJECTIVES

This work is based on the work of Benedetta Mennucci and José M. Martínez (ref. 10).Their work was motivated as a first part of a study of solvation in peptides using quantum-mechanical and classical approaches. In this study, the peptide is modelled as its simple analogue, namely, the N-methylacetamide, and the effects of the solvent (water) are introduced at three different levels: continuum description, using solute-solvent clusters, and using the same clusters embedded in an external continuum. Finally, a UV and IR spectra are calculated and analyzed.

Here, the same topics are followed, although molecular dynamics discussion is entirely obviated. Quantum-mechanical topics are adapted, usually reduced to the most stable conformation or expanded in level of theory and solvation modelling in order of carry out the following objectives:

• Reproduce Mennucci and Martínez results.

• Discuss about differences in the geometry of N-methylacetamide in vacuum and in solvation (water) for different basis sets and solvation models.

• Introduce a dispersion correction and evaluate its role in calculations.

• Analyze UV and IR spectra in vacuum and in solvation.

• Study the solvation effects on geometry, UV and IR spectra and provide a simple justification.

5.

C

OMPUTATIONAL DETAILS

Several calculations were performed to achieve the previous objectives. Here, it is presented how different inputs were elaborated in Gaussian 03 and Gaussian 09 (ref. 12 and 13).

5.1.

S

CANS AND GEOMETRY OPTIMIZATION

For all systems (single trans-NMA, cis-NMA and NMA clusters) both in vacuum and in solution, geometry optimizations were performed at the density functional theory using the

B3LYP hybrid functional. Two different basis set were used: 6-31+G(d,p) and 6-311++G(d,p). To the latter basis set Grimme's GD3BJ dispersion correction were added. IR spectra were computed in an optimization+IR calculation.

Scans of both methyl group were performed either in vacuum and in solution to probe which conformation is a true conformer and then, perform the geometry optimizations. This time, 6-31+G(d,p) basis set was used.

Also using 6-31+G(d,p) basis set, a scan of cis-trans conformations were executed in order to estimate the energetic barrier of this isomerisation.

For optimization and scan the convergence criteria is set to tight. Also, the grid used for DFT calculations is set ultrafine. Both criteria are needed in DFT to achieve good results.

5.2.

S

OLVATION

Two (actually, three) different solvation models were used for all systems. IEF-PCM solvation model were used in two different versions: Gaussian03 and Gaussian09. Although both versions uses the same modelling, they are slightly different. SMD solvation model were also used.

5.3.

UV

SPECTRA

To evaluate absorption energies, time-dependent DFT was used with B3LYP functional and the Dunning basis set with one diffuse (+) and one polarization (d) function on all O, N and C atoms (d95v+(d)). The same calculations were also performed for all system with a 6-311++G(d,p) basis set.

Calculations were performed with and without the Tamm-Dancoff approximation (TDA) to TD-DFT in order to test the quality of the approximation.